The
Schrodinger
Equation: The Schrodinger equation is the
foundation of the wave-particle formulation of quantum mechanics.
The Schrodinger equation cannot be derived; it was the
product of remarkable insight by Erwin Schrodinger, and has been
successful at explaining observed quantum measurements. The
Schrodinger equation takes the form:

H(psi)
= E(psi)

psi is called the
wavefunction - it is a function that describes the amplitude of a
wave at any position x.

H is a mathematical operator, or set of instructions about
what to do with the wavefunction. These instructions could
be "multiply by 2," "take the derivative," "take the second
derivative,"etc.

E is a constant
that describes the energy of the wavefunction psi
at any position x.

The Schrodinger equation is therefore a very special equation.
It says that the system is described by a function which, if
you do something to it like take the second derivative, you get
your exact function back, times a constant which represents the
energy of the wavefunction.

The wavefunction psi has
some interesting properties.

1) It must solve the Schrodinger equation.

2) It must be continuous.

3) The probability of the wavefunction, psi2,
must be normalized. This is to make sure that the system
described by the wavefunction allows for finding the system
somewhere in allowed space.

This last condition is particularly important and provides a
useful boundary condition for solving problems. It is
important to remember that this condition applies to the
probability of the wavefunction, not the wavefunction itself.

Particle-in-a-box:
The Schrodinger equation can be solved exactly for a handful
of problems. The simplest of these problems is called
particle-in-a-box, and is a classic example of the utility of the
Schrodinger equation.

Let's define a particle confined to a 1-dimensional box of length
L in which the potential
energy of the particle inside the box is zero, and outside the box
is infinite. Based on these conditions, we can find a
wavefunction psi and the
energy of the particle E.
By finding the correct wavefunction and solving for E,
we find that it is quantized automatically - no assumptions
required. Our solution to the particle-in-the-box is attached
here: